Abstract

The design and construction of Fresnel-type optics particularly suitable for use with area-type photoelectric receivers are described. Such optics have several advantages over conventional lens and mirror systems; large aperture systems may be constructed cheaply and are light in weight, the radiant energy can be distributed uniformly over the sensitive surface to avoid local saturation, and the optics can be economically custom-made for each application. The design procedure is explained and a method given for the construction of these Fresnel-type lenses from plastic in a modestly equipped machine shop. The design predictions and experimental performance data are compared for lenses designed for three different applications.

© 1951 Optical Society of America

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References

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  1. Miller, McLeod, and Sherwood, J. Opt. Soc. Am. 38, 1104(A) (1948); J. Opt. Soc, Am. 41, 807 (1951).

1948 (1)

Miller, McLeod, and Sherwood, J. Opt. Soc. Am. 38, 1104(A) (1948); J. Opt. Soc, Am. 41, 807 (1951).

McLeod,

Miller, McLeod, and Sherwood, J. Opt. Soc. Am. 38, 1104(A) (1948); J. Opt. Soc, Am. 41, 807 (1951).

Miller,

Miller, McLeod, and Sherwood, J. Opt. Soc. Am. 38, 1104(A) (1948); J. Opt. Soc, Am. 41, 807 (1951).

Sherwood,

Miller, McLeod, and Sherwood, J. Opt. Soc. Am. 38, 1104(A) (1948); J. Opt. Soc, Am. 41, 807 (1951).

J. Opt. Soc. Am. (1)

Miller, McLeod, and Sherwood, J. Opt. Soc. Am. 38, 1104(A) (1948); J. Opt. Soc, Am. 41, 807 (1951).

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Figures (11)

Fig. 1
Fig. 1

Comparison of lens types.

Fig. 2
Fig. 2

Geometrical considerations for a Fresnel-type lens with the steps on the back (Case I).

Fig. 3
Fig. 3

Geometrical considerations for a Fresnel-type lens with the steps on the front (Case II).

Fig. 4
Fig. 4

Interference by adjacent step when the steps are on the front of the lens.

Fig. 5
Fig. 5

Change of transmission with angle of incidence due to Fresnel reflection losses.

Fig. 6
Fig. 6

Field of view (a) as usually defined, (b) for no decrease in signal from photodetector, and (c) for moderate decrease in signal from photodetector.

Fig. 7
Fig. 7

Fresnel lenses described in this report.

Fig. 8
Fig. 8

Setting the angle of the compound prior to cutting the step.

Fig. 9
Fig. 9

The tool in cutting position.

Fig. 10
Fig. 10

Tool shapes used for machining plastic lenses.

Fig. 11
Fig. 11

Electrical response with various optics.

Tables (4)

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Table I Data on the 20-inch aperture lens ( 3 8 -inch steps).

Tables Icon

Table II Data on the test lens (6.25-inch aperture, 6-inch focal distance) and the small lens (3-inch aperture, 2.75-inch focal distance).

Tables Icon

Table III Transmission data on the 20-inch aperture lens (Case II).

Tables Icon

Table IV Transmission data on the test lens and the small lens.

Equations (36)

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n sin γ = sin β ,
γ = α             and             β = α + δ ,
n sin α = sin ( α + δ ) ;
δ = arc tan ( r / f.d. + 1 2 w tan α ) ;
n sin α = sin [ α + arc tan ( r / f.d. + 1 2 w tan α ) ] , arc sin ( n sin α ) = α + arc tan ( r / f.d. + 1 2 w tan α ) , r = ( f.d. + 1 2 w tan α ) tan × [ arc sin ( n sin α ) - α ] .
sin α = n sin β , n sin γ = sin δ ,
γ = α - β ;
n sin ( α - β ) = sin δ
sin δ = n ( sin α cos β -cos α sin β ) ;
sin β = ( sin α n )             and             β = arc sin ( sin α n ) ,
sin δ = n sin α cos [ arc sin ( sin α n ) ] -cos α sin α ;
x = 1 2 w tan α
y = t - 1 2 w tan α ,
z = y tan γ
z = ( t - 1 2 w tan α ) tan γ ;
γ = arc sin ( sin δ n ) ,
z = ( t - 1 2 w tan α ) tan [ arc sin ( sin δ n ) ]
s = r - z ,
tan δ = s f.d. = r - ( t - 1 2 w tan α ) tan { arc sin [ ( sin δ ) / n ] } f.d. .
r = f.d. tan arc sin { n sin α cos × [ arc sin ( sin α n ) ] -cos α sin α } + ( t - 1 2 w tan α ) tan { arc sin × [ sin α cos ( arc sin ( sin α n ) ) - cos α sin α n ] } .
F 0 = π r 2 E 0 = k π r 2 I 0 ,
F / F 0 = k π r 2 I / k π r 2 I 0 = I / I 0 ,
F F 0 = 1 A i = 1 n ( F F 0 ) i A i = 1 A i = 1 n ( I I 0 ) i A i ,
( F / F 0 ) i = K i ( I / I 0 ) i ,
F F 0 = 1 A i = 1 n ( F F 0 ) i A i = 1 A i = 1 n K i ( I I 0 ) i A i .
( I I 0 ) i = [ 1 - sin 2 ( φ 1 - φ 2 ) 2 sin 2 ( φ 1 + φ 2 ) - tan 2 ( φ 1 - φ 2 ) 2 tan 2 ( φ 1 + φ 2 ) ] i · [ 1 - sin 2 ( φ 3 - φ 4 ) 2 sin 2 ( φ 3 + φ 4 ) - tan 2 ( φ 3 - φ 4 ) 2 tan 2 ( φ 3 + φ 4 ) ] i
K i = 2 π ( r i + 1 2 c i ) ( w i - c i ) 2 π r i w i = ( r i + 1 2 c i ) ( w i - c i ) r i w i .
F F 0 = 1 A i = 1 n 2 π ( r i + c i 2 ) ( w i - c i ) ( I I 0 ) i .
a i = w i tan α i - 1 , [ sin ( 1 2 π - α i ) ] / b i = [ sin ( 1 2 π + β i ) ] / tan α i - 1 , b i = w i tan α i - 1 sin ( 1 2 π - α i ) / cos β i , c i = b i sin ( α i - β i ) = w i tan α i - 1 sin ( 1 2 π - α i ) sin ( α i - β i ) / cos β i ;
sin ( 1 2 π - α i ) = cos α i
sin ( α i - β i ) = sin α i cos β i - cos α i sin β i ,
sin ( α i - β i ) = sin α i cos [ arc sin ( sin α i n ) ] - cos α i sin α i n
c i = w i tan α i - 1 cos α i sin α i cos { arc sin [ ( sin α i ) / n ] × { cos [ arc sin ( sin α i n ) ] - cos α i n } .
F T = ( I s π r 2 / d 2 ) ( F / F 0 )
E = I s π r 1 2 d 2 ( F F 0 ) 1 π r 2 2 = r 1 2 I s r 2 2 d 2 ( F F 0 ) ,
1 2 π p 2 + q ( p 2 - q 2 ) 1 2 + p 2 arc sin ( q / p ) .